Limits and Intro to Derivatives

Bren Calculus Workshop


Nathaniel Grimes

Bren School of Environmental Science

Last updated: Sep 15, 2024

Team Review

  • How did everyone feel about the problem set?

  • Any questions?

  • Discuss with Team

Disclaimer


Functions are like baking receipes


  • Assemble all your ingredients (independent variables like \(x_1\), \(x_2\))

  • Follow the instructions to mix, bake, and decorate (Instructions of function)

  • End up with final product (\(f(x)\))

Typically we use the notation \(f(x)=x\), but we can always use different representations like \(g(x)=x\) or \(y=x\)

Properties of functions


For each combination of independent variables, there is exactly one value of the dependent variable.

  • \(f(1)\ne\{2,3\}\) i.e. if I put 1 into the function I can’t have both 2 and 3 come out

  • Vertical line test

Functions can be continuous or discontinuous

  • Continuous: For every value of x, f(x) returns a number

  • Discontinuous: Parts are undefined

Are these functions


Continuous

Discontinuous

Limits


Definition:

The value a function approaches as the input approaches a specific value

\[ \large \lim_{x\to c} f(x)=L \]

Verbally:

“The limit of the function f(x) as x approaches the value c is L”

Start with an example


  • What is the limit of \(f(x)=2x^2-4\) as x approaches 2?
  • Strategy to solve: Evaluate f(x) at x getting closer and closer to 2

  • Quick sanity check, what should \(f(2)=?\)

  • \(f(2)=4\)

From both directions it looks like f(x) converges to 4

\[ \lim_{x\to c}(2x^2-4)=4 \]

Graphical Solution


Finding limits of functions when f(x) is continuous at c is easy.

\[ \lim_{x\to c}f(x)=f(c) \]

What happens with discontinuous functions?


Try with your group the same approach for:

\[ \lim_{x\to 2}f(x)=\frac{|x-2|}{x-2} \]

Hint: Start very close to 2 like 1.9 and 2.1

We broke the universe


  • Dividing by zero is impossible

  • The function approaches different values from either side

  • Therefore…

Clearer on a graph


Technically the limit is “Undefined”, because it does approach a finite value. Does not exist happens when the limits spiral off to infinity in both directions (tan graph)

Defined Limit at an undefined point


  • Discontinuous functions can still have limits

  • Find the limit of \(f(x)=x+1,x\ne 2\) as x approaches 2

x=2 may not exist, but we can still find a limit because it consistently approaches 3 from both directions

Team Assessment

Work with your team to complete these task

Part 1:

\[ \lim_{x\to3}g(x)=0 \]

\[ \lim_{x\to-4}g(x)=3 \]

\[ \lim_{x\to 3} g(x)= \text{DNE} \]

Work with your team to complete these task

Part 2:

  1. Can you think of examples where discontinuous functions might exist in environmental science?

  2. Choose a a team to draw an example of one of these statements:

  • \(\lim_{x\to 4^-} f(x)\) and \(\lim_{x\to 4^+}f(x)\) are both infinite

  • \(\lim_{x\to 3} f(x)=2\), but \(f(3)=0\)

  • \(\lim_{x\to 5^-} f(x)=4\) and \(\lim_{x\to 5^+} f(x)=2\)

  • \(\lim_{x\to -3} f(x)=-5\) but \(f(-3)=-5\)

Introduction to Derivatives

Putting it all together


Recall average rate of change and instantaneous

Taking the average rate of change to a set limit will eventually converge to the instantaneous.

Walk Trhough Example


Walk Through Example


Walk Through Example


Walk Through Example


Walk Through Example


Walkthrough


Tangent Lines


What if we set \(\Delta x=0\)? Then we would have a slope line that only touches our function at exactly \(x\).

These are called tangent lines

Put our example in math notation


\[\text{slope}=\frac{y_2-y_1}{x_2-x_1}\]

Choose a point along the function \((x,f(x))\)

Choose a different point on the function \(\Delta x\) away \((x+\Delta x,f(x+\Delta x))\)

Add these into the slope equation

\[ \text{slope}=\frac{f(x+\Delta x)-f(x)}{(x+\Delta x) -x}=\frac{f(x+\Delta x)-f(x)}{\Delta x} \]

Derivative Definition


\[ \large f'(x)=\lim_{\Delta x \to 0}=\frac{f(x+\Delta x)-f(x)}{\Delta x} \]


Most common notation

\[ \large f'(x)\text{, or }\frac{dy}{dx} \]

Not all functions are differentiable


  • All differentiable functions are continuous, but not all continuous functions are differentiable

  • The absolute value function is one example \(y=|x|\)

Calculus and Derivatives are the study of change


Environmental Science is also a study of change


Environmental Science is also a study of change


Environmental Science is also a study of change


Rules for Differentiation


Constant Rule

  • If \(f(x)\) is contant, then for all \(x\), \(f'(x)=0\)

\[ \begin{align} &y=a &\frac{dy}{dx}=0 \end{align} \]

Power Rule

\[ \frac{d}{dx}[x^n] =nx^{n-1} \]

Examples


\[ \begin{align} &y=100 & &y=5x^5 & &y=\frac{1}{x^2} \end{align} \]

Rules for Differentiation


Sum and Difference Rules

\[ \begin{align} \frac{d}{dx}&=[f(x)+g(x)]=f'(x)+g'(x) \\ \frac{d}{dx}&=[f(x)-g(x)]=f'(x)-g'(x) \end{align} \]

Looks really scary. All it says, if the function has pieces that are added or subtracted you can take the derivative of each individual piece.

Example


\[ \begin{align} &y=x^2+8x+4 & &y=x^3-x^2+x-15 \end{align} \]

Team Assessment

  1. As a team, list 5 fields of environmental science where studying the rate of change and derivatives would be important

  2. Which rules should you use to take these derivatives?

\[ \begin{align} \text{A) }& f(x)=3x^4 &\text{B) } y=4x^2+3x-16 \end{align} \]

  1. Find the derivatives of these functions

\[ \begin{align} &\text{A) } y=3x^2 & &\text{B) }h(x)=y=7x+4 & &\text{C) }g(y)=\sqrt{y} \end{align} \]

  1. Discuss why derivatives need continuity. It might help to think about why some continuous functions don’t have derivatives.